Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C228 |
| direct product | C27×D3 |
| unambiguous semidirect product | C3⋊C28 |
| unambiguous semidirect product | (C2×C3)⋊C27 |
| Fitting, socle | C27×C3 ⋊ C2 |
| derived, Sylow | C3 ⋊ C28 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C27 | D3 |
| commutator, cosocle, hypocenter | C3 | C28 |
| Frattini | C1 | G |
| Fitting, socle | C27×C3 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C27 | |
834398 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
384 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083472);